ForProject Whack a Mole I need a 32.768kHz crystal oscillator. I found this circuits on the Interwebs and gave it a try:

It wouldn’t go. I messed about changing component values for while, then decided to actually try to understand the circuit. Now for an oscillator to work, we need an amplifier with a gain of greater than 1, and a phase shift of 360 degrees to get positive feedback.

The circuit above is an amplifier, with the crystal network connected between the collector output and base input. We get half of the 360 degree phase shift by using a common emitter topology, which is an inverting amplifier. So the crystal network must provide the other 180 degrees. On a good day. If it’s working.

First problem – the transistor was saturated, with Vc stuck near 0V. For an oscillator to start noise gets amplified, filtered by the crystal, amplified again etc. I reasoned that if the amplifier wasn’t biased to be linear, the oscillations couldn’t build up. So I reduced the collector resistor to 6.8k, and changed the the base bias resistor to 1.8M to get the collector voltage into a linear region. So now we have Vc=3.2V with a 5V supply.

But it still wouldn’t go. On a whim I adjusted the supply voltage up and then down and found it would start with a supply voltage beneath 3V, but not any higher. Huh?

Much fiddling with pencil and paper followed. Time for a LT Spice simulation of the “AC model” of the circuit:

I’ve “opened the loop”, to model the collector driving the crystal network which then drives the base impedance. On the left is a voltage source and 6.8k resistor that represents the collector driving the 330k resistor and an equivalent model of the crystal.

The values Lm, Cm, Rm, are the “motional” parameters. They are what the mechanical properties of the crystal look like to this circuit. The values are amazing, unrealizable if you are used to regular electronic parts. I found Cm = 1fF (1E-15 Farads, or 0.001 pF) in a 32kHz crystal data sheet, then solved f=1/(2*pi*sqrt(LC)) for Lm to get the remarkable value of 24,000 Henrys. Wow.

#### Phase Shift

With Vcc=5V, we have Vc=3.2V, so a collector current Ic = (5-3.2)/6800 = 0.265mA. I’m using a small signal transistor model with the emitter resistance re=26/Ic = 26/0.265 = 100 ohms. The effective impedance looking into the base rb=beta*re = 100*100 = 10k ohms (2N3904’s have a minimum beta of 100).

OK, so here is the phase response near 32kHz:

Well it looks about right, a phase shift of 170 degrees, which is close to the target of 180 degrees.

Now, can we explain why the oscillator starts with a reduced supply voltage? Well, reducing Vcc would reduce Ic and hence increase rb, the base impedance the crystal network is driving. So lets double rb to 20k and see what happens to the phase:

It gets closer to 180 degrees! Wow, that means it is more likely to oscillate. Just like the actual circuit.

So – can I induce it to oscillate on a 5V supply? Setting rb back to 10k, I messed about with C1 and C2. Increasing them to 82pF moved the phase shift to just on 180 degrees. I soldered 82pF capacitors into the circuit and it started on a 5V rail. Yayyyyy. Go Spice simulations.

#### Loop Gain

But what about the loop gain? Well here is the magnitude plot near 32kHz:

The maximum gain is -22dB at series resonance, followed by a minimum gain at parallel resonance. We need a net gain around the loop of 1 or 0dB. So the gain of the amplifier must be at least +22dB to get a net gain of 0dB around the loop.

The gain of common emitter amplifier is Rc/re = 20*log10(6800/100) = 36dB. So we have enough gain. At the reduced supply voltage let say Ic is halved, so re doubles. This would reduce the loop gain to 30dB. However rb=beta*re would also double to 20k. Spice tells me the maximum gain of the crystal network is now -16dB, as rb=20k loads the circuit less. So still plenty of margin for oscillation – which is what happens in the real hardware.

Increasing C1 and C2 to 82pF produced a crystal network gain of -24dB. With a 5V supply the amplifier gain is 36dB so we have enough loop gain to make this puppy oscillate. Which it does, eventually. It takes about 10 seconds for the oscillations on the collector to hit the supply rails. From some reading I understand a slow start in the order of seconds up is common for these oscillators.

Matt, VK5ZM, suggested the function of the 330k resistor is to limit the power through the crystal. These tiny crystals are rated at just 1uW maximum power. With 1Vrms AC drive, Spice measured a current of 7.2uA through the crystal series resistance Rm=35k at the resonant frequency, which is a power of 35E3*(7.2E-6)^2 = 1.8uW. Oops, a bit much. However I think increasing the 330k resistor might reduce the loop gain. And I have a big bag of spare crystals.

Matt also pointed out there are some parasitic capacitances from the transistor that could be included in the model.

Here is the final circuit, that works on 5V:

#### Links

Open Loop LT Spice simulation of the crystal oscillator network.

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